Rigidity of lattice actions on boundaries

A uniform lattice in SL_n(R) with n at least 3 does not admit non-trivial deformations in SL_n(R), thanks to Margulis super-rigidity. However, such a lattice has a natural action on the Furstenberg boundary. Then, a natural question is whether the lattice can be deformed in this (huge) homeomorphism group of the Furstenberg boundary. We prove a semi-conjugacy rigidity result for such deformations. Surprisingly, we also show that no such rigidity holds for the visual boundary of the symmetric space SL_n(R)/SO_n. This is joint work with Chris Connell, Thang Nguyen, and Ralf Spatzier.